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A fast algorithm for computing the Boys function

Gregory Beylkin, Sandeep Sharma

TL;DR

The paper tackles the efficient evaluation of the Boys function $F(n,z)$ for real and complex $z$, a key ingredient in Gaussian integral computations. It introduces two complementary strategies based on nonlinear exponential approximations: (i) a quadrature-based approach for $F(0,z)$ with complex arguments and $\mathcal{R}e(z)\ge0$, and (ii) a near-optimal exponential approximation for $g_n(s)=(1-s)^{n-1/2}$ to compute $F(n_{\max},z)$ (with recurrences to obtain all lower $n$) for both real and complex $z$, along with explicit error bounds and offline coefficient computation. The methods yield tight accuracy with manageable computational costs and are amenable to parallelization, demonstrated with Fortran 90 implementations and supplementary material for practical use. These contributions offer a robust, scalable framework for computing the Boys function in complex-argument scenarios, broadening its applicability in quantum chemistry and related scattering problems.

Abstract

We present a new fast algorithm for computing the Boys function using a nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.

A fast algorithm for computing the Boys function

TL;DR

The paper tackles the efficient evaluation of the Boys function for real and complex , a key ingredient in Gaussian integral computations. It introduces two complementary strategies based on nonlinear exponential approximations: (i) a quadrature-based approach for with complex arguments and , and (ii) a near-optimal exponential approximation for to compute (with recurrences to obtain all lower ) for both real and complex , along with explicit error bounds and offline coefficient computation. The methods yield tight accuracy with manageable computational costs and are amenable to parallelization, demonstrated with Fortran 90 implementations and supplementary material for practical use. These contributions offer a robust, scalable framework for computing the Boys function in complex-argument scenarios, broadening its applicability in quantum chemistry and related scattering problems.

Abstract

We present a new fast algorithm for computing the Boys function using a nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.

Paper Structure

This paper contains 11 sections, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Approximation of $F\left(0,z\right)$ for complex valued argument
  3. The case $\mathcal{R}e\left(z\right)\ge0$.
  4. The case $\mathcal{R}e\left(z\right)<0$
  5. Approximation of $F\left(n_{\max},z\right)$ for real and complex arguments
  6. Implementation details
  7. Conclusion
  8. Supplementary Material
  9. Acknowledgements
  10. Data availability
  11. Appendix: Construction of quadrature in \ref{['EQ:QUAD']}

Figures (1)

  • Figure 1: The function $g_{12}\left(s\right)$ in \ref{['eq:function g']} and the error of its near optimal approximation via exponentials in \ref{['eq:key approximation']} with parameters described in Table \ref{['tab:Weights-and-exponents']}.